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Full text of "Physics 101, Black Holes and Gravitational Fields Assignment"

Physics o P Black Holes 



Patrick Bruskiewich 



June, 2002 



t 



970 



CHAPTER 



unitization 




Figure 1 

How the geometry of space is 
warped by a spherical (nonro- 
tating) black hole. The radial 
distance from any point to the 
center of the black hole can be 
measured in two ways. One 
way is to measure ihe circum- 
ference of a circle passing 
through the point and cen- 
tered on the hole, and divide 
by 2tt. A second way is to 
measure the distance from the 
point to the center along a 
radial line on the curved sur- 
face. These two measures will 
be nearly equal far from the 
hole where space is "flat," 
but highly discrepant near 
the black-hole horizon where 
the funnel narrows to a ver- 
tical tube. 



Black Holes 

Alan P. Lightman 
Cornell University 



Imagine a region of space where attractive gravitational force is so intense that 
light rays venturing too near can be bent into circular orbit -a region from 
which no matter, radiation, or communication of any kind can ever escape. 
Such is a black hole, one of the most exciting creatures of theoretical physics 
and possibly the most bizarre object of space. 

Although they were implicitly predicted by Einstein's 1915 theory of grav- 
ity, general relativity, black holes were first theoretically "discovered" by Op- 
penheimer and Snyder in 1939. Because of their highly nonintuitive proper- 
ties, however, black holes were not taken seriously by most physicists and 
astronomers until the mid-1960s. We are now on the verge of confirming the 
discovery in space of the first black hole. 

The concept of a black hole stretches our ideas of time and space to their 
limits. The surface of a black hole, called its horizon, is a closed boundary 
within which the escape velocity exceeds the velocity of light. The prediction 
of such a surface for sufficiently compact bodies can be made just on the basis 
of Newton's theory of gravity together with special relativity: the escape veloc- 
ity of a particle launched from the surface of a spherical mass M of radius R is 
v e = V2GM/R (see Chapter 9). When MIR satisfies 2GM/R > c 2 , v e exceeds the 
speed of light and no particle or photon may escape, as required by special rel- 
ativity. As a remarkable result, the interior of a black hole is causally discon- 
nected from the rest of the universe; no physical process occurring inside the 
horizon can communicate its existence or effects to the outside. For a spherical 
black hole of mass M, the horizon is a sphere of circumference equal to 2ir 
times the Schwarzschild radius of the hole Re, where Re = 2GM/c 2 (exact nu- 
merical coincidence of this radius with the newtonian analog above is ac- 
cidental). A black hole with a mass equal to that of our sun would have a 
Schwarzschild radius of 2.95 km. 

According to general relativity, time and space are warped by the gravita- 
tional field of massive bodies, with the warping most severe near a biack hole. 
Gravity affects all physical systems in a universal way so that all clocks 
(whether transitions of an ammonia molecule or heartbeats of a human being) 
and all rulers would indicate that time is slowed down and space stretched out 
near a black hole (see Figure 1). Alternatively, one can describe the effects of 
the hole's gravitational field on a local measurement of time and space in-' 
tervals as an acceleration of the local Lorentz frame (in which special relativity 
is valid) with respect to other local Lorentz frames at different locations. 

Black holes are formed when massive stars undergo total gravitational 
collapse. While emitting heat and light into space, stars balance themselves 
against their own gravity with the outward thermal pressure force generated 
by heat from nuclear reactions in their deep interiors. But every star must die. 
When its nuclear fuel is exhausted, a star will contract. If its mass is Jess fen 
about three times the mass of our sun, the shrinking star will stabilize at a 
smaller size when the inward pull of gravity cannot force the star's constituent 
particles any closer together. Such a star will spend eternity as a white dwarf or 
neutron star. But if the star's mass exceeds about 3 solar masses, theory in- 
dicates that no amount of outward pressure force can stave off the overwhelm- 
ing crush of gravity and the star will plunge in upon itself, disappearing for- 
ever from sight and giving birth to a black hole. 

Recent mathematical proofs, using Einstein's general relativity theory, dem- 
onstrate that a black hole is one of the simplest objects of nature and can be 
completely characterized by only three quantities: its mass, angular momen- 
tum, and net electrical charge. Other than these three quantities, all informa- 
tion of the progenitor star, e.g., whether it was composed of particles or 
antiparticles, whether it was pancake-shaped or spherical, is lost via gravita- 





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REVIEW 



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tonal and electromagnetic waves shortly after formation of the hole. Rotating 

sur o, JTk C f h °u S (n ° nrotat!n g hol « ^ "lied SchwarzschM holes), 

surround themselves with a reg,on called the ergasphere in which space and 

■me are so strongly twisted that all particles, photons, and even local Lorentz 

frames are forced to rotate about the hole. Lorentz 

ppn A ' fTu 6iStanCe ! fr ° m a b ' aCk h ° !e ' ' tS 8 ravl ^tional field behaves as if 
generated by an ordinary star of mass M, obeying the newtonian inverse- 
square law. Close to a black hole, the gravitational field is far stronger than 

mas" blar k P h1 1Cted ^ ^^^ theorv - A ™" being sucked into a solar- 
mass black hole would be npped to shreds by the differential force of gravitv 
across his body long before he reached the horizon 

Once the hole's horizon has formed we cannot receive any information 

lap" r dial dT^ ate °u ^ ?*«*"* "" ^ C ^°™ ° f ^ col 

ume andTnr 6 > A ff' ^^ that Ae Star 1S Crushed *> «"> vol- 
ume and mfmite dens.ty at the center of the black hole, forming there a point 

of infinite gravitational force called. a singularity. For a collapsing star of a few 

olar masses, the final stellar death throes would be over in a few hundred 

la°ss^ t 3 f C ° nd ( f S meaSUr6d l0Cally) - Quantum effects ' omi «ed in the 
classical theory of general relativity, could possibly halt the stellar collapse at 

onsH T 8 7 t K " Sity P ~ C '' hGi ~ 5 X 10 " S m/Cm: ' Where * > s Pick's 

prevent thTfc ^"Tu, ?*? ," ° f sin 8 uIa » tle ^ but such effects could not 
prevent the formation of black holes. 

, nn C ";, ent th J m ! eS u ° f Stellar evo,utl0n suggest that there could be as many as 
100 million black holes in our galaxy. The search for these black holes is not 
easy We could never detect a tiny black speck a few miles across against the 
mght sky. Instead we must search for signs of black holes interacting with 
their astrophysical neighborhoods. Most vulnerable to discovery would be a 
black hole orbiting a normal star in a binary system. Using Kepler's laws, anal- 
ysis of the magnitude and period of the doppler shift of the visible normal star 
a lows us to compute whether the unseen companion is massive enough to be a 
black hole In addition we might observe intense,, flickering x-rays produced 
when gas from the normal star is sucked toward the black hole "and heated to 
a billion degrees in its spiraling path to destruction in the hole. Such are the 

o a nTk , k bmary X " aV S ° UrCe CygnuS X- 1 ' an e * ceil <^ «mdidate 
for a black hole about 8000 light-years from earth in the constellation Cvgnus 

Black holes are a fundamental phenomenon of nature. Space may be littered 
with black holes tempting us with their secrets of time and space behind a 
cloak of impenetrable darkness, a challenge and a prize for the persistence of 
astronomers and physicists. 






Name: 



Black Hole Test 



Per 



Part A: True/False 




For each of the following, circle the T if the statement is true. If the statement is false, 
circle the F, and rewrite as a true statement on the line beneath. 

T F 1. Light can escape from a black hole. 



T F 2. Black holes were theoretically discovered by Oppenheimer 

and Snyder in 1939. 



T F 3. 



The theory used to predict black holes is Einstein's Theory of 
General Relativity. 



T F 4. Clocks go faster near a black hole. 



T F 5. Distances are shortened near a black hole. 



T F 6. A black hole can be characterized by only three quantities: mass, 

angular momentum and net electrical charge. 



T F 7. By looking at a black hole you can tell how it was formed. 



T F 8. 



A black hole the mass of the sun would have a Schwarzschild 
radius of 2.95 metres. 




Part B : Matching 

In the space to the left, write the letter of the description that best fits each term. 



Term : 
.9. Schwarzschild hole 

.10. A Black Hole 

.11. Ergosphere 

.12. Kerr hole 
.13. Horizon 

.14. Light 
.15. Singularity 
.16. Space 



Description : 

A. an open boundary within which the escape 
velocity exceeds the velocity of light. 

B. venturing too near a black hole can be bent 

into circular orbit. 

C. point of infinite density at the centre of 
a black hole 

D. may be littered with black holes. 

E. are formed when massive stars undergo 
total gravitational collapse. 

F. region around a stationary black hole. 

G. a non-rotating black hole. 

H. a rotating black hole. 

I. a closed boundary within which the escape 

velocity exceeds the velocity of light. 

J. region around a spinning black hole. 




Part C : Multiple Choice 

In each of the following questions, put your answer in the space provided. 

17. The theory that implicitly predicts the existence of black holes is? 17, 



A. the special theory of relativity 

B. the general theory of relativity 



C. the theory of singularities 

D. the theory of stellar evolution 



18. Stars balance themselves against their collapse? 



18 



A. by gravitational force C. 

B. by spinning very fast D. 



by thermal pressure 
by nuclear forces 



19. A star will collapse into a black hole when 



19. 



A. its mass is less than three solar masses 

B. it is spinning at three times that of the sun 

C. its mass is greater than three solar masses 

D. it starts to become a white dwarf or neutron star 



20. 



Once the black hole's horizon has been formed 



20. 



A. we cannot receive any information from within the black hole 

B. its mass is crushed to zero volume at the centre 

C. its mass is crushed to infinite density at the centre 

D. all of the above 



21. 



To detect a black hole 



21. 



A. we can use a telescope to see then directly 

B. we cannot see them with a telescope 

C. we need to watch for the Doppler shift of a black hole 

D. we need to watch for the Doppler shift of a companion star 



22. 



The interior of a black hole 



22. 



A. is causally connected with the outside world 

B. is causually disconnected with the outside world 

C. is visible to a distant observer using an optical telescope 

D. is visible to a distant observer using a radio telescope 



23. At great distance from a black hole 



23. 



A. space is stretched or warped 

B. time slows down 

C. space behaves as if generated by a normal star 

D. we don't see or feel the black hole 



24. A man falling into a black hole 



24. 



E. would not feel a thing 

F. would find his watch going slower 

G. would be ripped to shreds 

H. would feel like he was be compressed 




Part D : Short Answers 

25. Using the principle of conservation of energy, prove that the radius of a 
black hole of mass M is the Schwarzschild Radius Rs = 2 GM/c 2 . (4 marks) 



26. Calculate the Schwarzschild Radius of the earth in metres. (4 marks) 



mass of the earth = 5.98 x 10 24 kg 

c = 3x10 8 m/s 

G = 6.67x1CT 11 Nm 2 /kg 2 



27. Describe how you might use a telescope to find a black hole 

(100 words or less) (4 marks) 




. 



28. In a gravitational field the frequency of light changes as light climbs out of the 
gravitational well. Starting with the expression for the change in gravitational 
energy U = mgz where z is the height above the star and using the Planck 
expression E = h v, where v is the frequency of the light find an expression for 
the change in frequency of the emitted light. (6 marks) 



29. Estimate the fractional change in frequency for light that climbs 10 metres 
from the surface of the earth. (4 marks) 



30. Given the expression for the fractional change in frequency for light that 
climbs out of a gravitational well, speculate what happens to the light from a star 
that begins to collapse into a black hole. ( 100 words or less) (4 marks)